Triality and its uses.

After the resuscitation of triality in Baez's commentary of Garrett Lisi work, it seems worthwhile to open a separate thread to discuss the expectatives of using triality to build the models Beyond Standard Model.

To start, I have found this one:

http://arxiv.org/abs/hep-th/0104050 Lee Smolin The exceptional Jordan algebra and the matrix string. It was a bold speculative paper from Lee, naming Baez in the last page and quoting Motl in the references, so perhaps it was the missed opportunity for the internet physicists to become an independent workforce. Tony invokes this paper a couple times in a closed thread at Woit's

Another example, SU(3) [tex]\supset[/tex] SU(2) x U(1), is recalled by T. Pengpan, P. Ramond (funny, I believed to remember it was Polchinski) in http://arxiv.org/abs/hep-th/9808190

Basically, the embedding of SU(2)xU(1) in SU(3) is to put a 1x1 and a 2x2 boxes inside a 3x3 box. In matrix form, there are three ways to do it: the two ones you can see pictorialy, and the one using the corner of the matrix:

It seems there is a rule "quaternionist" + "composites"+ "generations" = "octonian" but Adler is a counterexample. He concentrates a lot in quaternion theory but he does not hurry to generalize to octonions, nor to disgress on triality. Hmm.

H Georgi book, Lie Algebras in Particle Physics, was published in 1982 and it mentions all the issues but in very neutral way. The word "triality" is not invoked, but a full subchapter is titled "Fun with SO(8)".

There is also a last chapter that the first edition titles equal to the whole book (thus untitled) and the second one, even more neutral, refers as "odds and ends", or something so. It containts a subchapter Exceptional Algebras, which speaks of "octonians", and a short subchapter on Anomalies, that mentions the special case of SO(6) as being the only non anomaly free of its series (because it is equivalent to SU(4) and all the SU(n >=3 ) are not anomaly-free.).

The above is over my head but it is supposedly connected in some way to the things I play with.

By the way, I've just learned that wordpress (i.e. see http://carlbrannen.wordpress.com/ ) allows LaTex in its blogs and comments. They also allow private blogs where one can have more than one poster (by invitation only). The whole thing seems perfect for doing research. The only thing I could ask for beyond what they have is a method of making posts invisible on a post by post basis. Hmmm.

Another example, SU(3) [tex]\supset[/tex] SU(2) x U(1), is recalled by T. Pengpan, P. Ramond (funny, I believed to remember it was Polchinski) in http://arxiv.org/abs/hep-th/9808190

Basically, the embedding of SU(2)xU(1) in SU(3) is to put a 1x1 and a 2x2 boxes inside a 3x3 box. In matrix form, there are three ways to do it: the two ones you can see pictorialy, and the one using the corner of the matrix:

It's a quick check to see these matrices satisfy P^2=P and tr(P)=1, hence are primitive idempotents. These idempotents form an (orthonormal) projective basis for [tex]\mathbb{CP}^2[/tex], and each SU(2)xU(1) embedding inside SU(3) will leave one of the [tex]\mathbb{CP}^2[/tex] basis idempotents invariant, while transforming the other two. By combining two or more different SU(2)xU(1) embeddings, i.e. UVPV*U*, we recover a general SU(3) transformation, much like we do for Euler angles and SO(3). So triality, in the sense of embeddings of maximal subgroups, can be geometrically interpreted as the three ways we can transform the basis of a projective plane.

This construction also works for the higher projective planes [tex]\mathbb{HP}^2[/tex] and [tex]\mathbb{OP}^2[/tex], where the corresponding subgroups are Sp(4)xSp(2)~SO(5)xSO(3) and SO(9) for the groups Sp(6) and F4, respectively. The SO(9) case is relevant to Smolin's model, and more recently, to BPS black holes (hep-th/0512296).

More practically, the primitive idempotent construction is related to Carl's Lepton Masses work, where each lepton generation corresponds to a different projective basis element for [tex]\mathbb{CP}^2[/tex] and the masses to their corresponding eigenvalues. Therefore, using Carl's circulant Hermitian matrix construction, it can be shown that triality and SU(2)xU(1) embeddings in SU(3) map the three lepton generations to the projective basis for [tex]\mathbb{CP}^2[/tex].

There is some triality in the ways to embed a SU(2) inside SU(3), but also in the ways to embed SU(3) in SO(8). Note that you can not embed the full [LR] standard model group into SO(8), same that you can not embed SU(3) O----O into SO(5) O====O.